State-merging

In quantum information theory, quantum state merging{{cite journal|arxiv=quant-ph/0505062 | last1= Horodecki | first1=M.|last2=Oppenheim | first2=J.| last3=Winter | first3=A. |title=Partial quantum information|year=2005|journal=Nature|volume=436|pages=673–676 | doi = 10.1038/nature03909|issue=7051|pmid=16079840|bibcode=2005Natur.436..673H| s2cid= 4413693 }}{{cite journal|arxiv=quant-ph/0512247 |doi=10.1007/s00220-006-0118-x| last1=Horodecki | first1=M.|last2=Oppenheim | first2=J.| last3=Winter | first3=A. |title=Quantum state merging and negative information|year=2007|journal=Communications in Mathematical Physics|volume=269|issue=1 |pages=107–136|bibcode=2007CMaPh.269..107H|s2cid=119421959}} is the transfer of a quantum state when the receiver already has part of the state. The process optimally transfers partial information using entanglement and classical communication. It allows for sending information using an amount of entanglement given by the conditional quantum entropy, $H(A|B)\backslash ,=\backslash ,H(AB)-H(B)\; \backslash ,\; .$ with $H(A)$ the Von Neumann entropy, $H(A):=-Tr\backslash rho\_A\backslash log\backslash rho\_A$. It thus provides an operational meaning to this quantity.

Unlike its classical counterpart, the quantum conditional entropy can be negative. In this case, the sender can transfer the state to the receiver using no entanglement, and as an added bonus, this amount of entanglement can be gained, rather than used. Thus quantum information can be negative.

The amount of classical information needed is the mutual information $I(A:R):=H(A)+H(R)-H(AR)$. The case where the classical communication is replaced by quantum communication was considered in.{{cite journal|arxiv=quant-ph/0606225 |doi=10.1098/rspa.2009.0202 |first1=A.| last1= Abeyesinghe |first2=I.|last2=Devetak|first3=P|last3=Hayden|first4=A.|last4=Winter

|title=The mother of all protocols: restructuring quantum information's family tree|journal=Proc. R. Soc. A | volume=465 |pages=2537–2563|year=2009|issue=2108 |bibcode=2009RSPSA.465.2537A|s2cid=44021187 }} This is known as the Fully Quantum Slepian-Wolf Theorem, since everything is sent down the quantum channel. A single-shot version of state merging was found by Berta,{{cite arXiv|eprint=0912.4495 | last1= Berta|first1=M.|title=Single-shot quantum state merging|year=2009|class=quant-ph}} and a multiparty single shot version was found in.{{cite arXiv|eprint=1011.1974 | last1=Dutil|first1=N. | last2=Hayden | first2=P.|title=One-shot Multiparty State Merging|year=2010|class=quant-ph}} The quantum discord has been interpreted using state merging.{{cite journal | title=Interpreting quantum discord through quantum state merging |last1=Madhok |first1=V. |last2=Datta|first2=A.|year=2011 | doi=10.1103/PhysRevA.83.032323 | journal=Physical Review A | volume=83 | issue=3|page=032323 |arxiv=1008.4135|bibcode=2011PhRvA..83c2323M|s2cid=10898479 }}